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Properties

Label 2-777-777.137-c0-0-1
Degree $2$
Conductor $777$
Sign $0.308 + 0.951i$
Analytic cond. $0.387773$
Root an. cond. $0.622714$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·5-s + (0.5 + 0.866i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.499i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s − i·17-s − 19-s + (−0.866 + 0.5i)20-s + (0.866 + 0.499i)21-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·5-s + (0.5 + 0.866i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.499i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s − i·17-s − 19-s + (−0.866 + 0.5i)20-s + (0.866 + 0.499i)21-s + (−0.866 + 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(0.387773\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :0),\ 0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193089573\)
\(L(\frac12)\) \(\approx\) \(1.193089573\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.814507121800157929062062655180, −9.396120436087311194012884290123, −8.778329035743729947014642620780, −8.061887194566809577099452238082, −6.83944522431213588561024866733, −5.93612941703571267395545897992, −4.76106850912257751093378117841, −4.20354994606331893837615658952, −2.30256241384155093179851325914, −1.42568867514748221169517782317, 2.23493420133551506232438919887, 3.58546879383177297700089851918, 3.82460172966955993103133495167, 5.03714751571381716756015763228, 6.61475763971654735481094605609, 7.43312080704445551455470757173, 8.207228390529592382510824498875, 8.787761744066372904461225669273, 9.886565742833348513967088940467, 10.65294210123054719488750450042

Graph of the $Z$-function along the critical line